Suppose $X$ is a projective smooth surface, $C$ is a irreducible curve on $X$ with $C^2=0$ and $C\cdot K_X=-2$, then I saw on some proofs of classification of extremal ray on surfaces that $|nC|$ is base point free for $n>>0$.
I can’t see why from the conditions we have. Could anyone give me an explanation.
The conditions imply that $C$ is a smooth rational curve by genus formula. We have an exact sequence $0\to\mathcal{O}_X((n-1)C)\to \mathcal{O}_X(nC)\to \mathcal{O}_C\to 0$ for any $n$. Thus, we have a surjection $H^1((n-1)C)\to H^1(nC)$ for all $n$. Since these are finite dimensional, we see that for all large $n$, this surjection is an isomorphism. Thus, the map $H^0(nC)\to H^0(\mathcal{O}_C)$ is a surjection for all large $n$. This immediately implies that $nC$ is base point free for all large $n$.